Cross-numeraire relative pricing of ethereum: data authority, market frictions, and static arbitrage bounds

Abstract Using Gate.io hourly spot-order-book archives from January 2022 through June 2023, this paper reconstructs a long-horizon five-second panel for comparing the direct Ether/Tether (ETH/USDT) quote with the synthetic Ether/Bitcoin (ETH/BTC) $$\times $$ Bitcoin/Tether (BTC/USDT) representation. The reconstruction preserves ordered replay, source time, valid best-bid-and-offer states, and explicit missingness. Of 39,312 expected market-hour files, 39,147 are available, but only 4,391,146 of 9,434,880 grid timestamps (46.54%) support all three legs. All pricing distributions are conditional on joint archive readiness. On those timestamps, the median signed direct–synthetic wedge is 0.164 basis points and the median absolute wedge is 1.416 basis points; the ETH/BTC leg contributes approximately 91% of aggregate quoted friction. Gross directed inequalities occur in 20.25% of forward and 18.61% of reverse observations. Under a hypothetical five-basis-point fee per leg, only 64 forward and 56 reverse observations remain positive. Common-support diagnostics show that zero-ready hours have larger observed ETH/USDT absolute returns, realized volatility, quoted spreads, and archived message counts, but price-state coverage is only 56 of 160 such hours and no missing-at-random claim is made. A secondary source-age-bounded Q2 2024 Gate.io–OKX comparison also leaves a nearly empty after-cost boundary. These archived quotes identify fee-sensitive static arbitrage bounds, not completed three-leg trades or realized trader-specific returns.